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On the non-linear buckling and post-buckling analysis of thin elastic shells
0020.7462lU453.M) Rrgamon
+ .W Press Ltd.
ON THE NON-LINEAR BUCKLING AND POSTBUCKLING ANALYSIS OF THIN ELASTIC SHELLS? H. Ruhr-Universitlt
STUMPF
Bochum, Lehrstuhl fur Mechanik 11, Universit6tsstraBe lSOlA, D-4630 Bochum I, Federal Republic of Germany (Receiued 3 June 1983)
Abstract-ln this paper the non-linear stability and post-buckling analysis of elastic structures is considered in the frame of a geometrically non-linear shell theory with moderate rotations. Using a total Lagrangian description variational statements as well as associated sets of shell equations including boundary conditions are derived to determine the fundamental quilibrium path, critical points of snap-through or bifurcation bucklingand also the post-buckling deformations. To present these equations in a compact form a unified operator description is introduced. It also allows one to prove some important properties, which are needed to construct appropriate approximation procedures like finite element methods. A shell example is calculated numerically by using a simple Rayleigh-Ritz approximation. It is shown that for a two-parameter loading the collection of critical snap-through buckling points is a catastrophe of the *cusp’ type.
I. INTRODUCTION
Basic contributions to the stability and post-buckling analysis of elastic shell structures had been given by Koiter in several papers, among which [l-3] are mentioned here. Using a general energy functional Budiansky investigated the theory of buckling and post-buckling ofelastic structures [4], where some applications are also given. In the literature the stability behaviour of many special shell structures has been the subject of a vast numer of further publications. The aim of this paper is to consider the buckling and post-buckling of elastic structures in the frame of a general non-linear elastic shell theory for which we use the energy-consistent so-called ‘moderate rotation shell theory’, characterized by the assumption of small strains and moderate rotations [5]. This theory leads to an adequate description of a wide area of shell problems of engineering interest and contains many simplified theories as those connected with the names of Sanders [6], Koiter [7] and Donnell-Marguerre-Vlasov. It is shown in [8] that only for one-dimensional problems as beams and columns and some very special shell structures in the post-buckling range a ‘large rotation shell theory’ [9, lo] can lead to remarkable different results. For the moderate rotation shell theory variational principles have been derived in [ 11, 123, while the Lagrangian stability equations are given in [13,14]. In this paper variational statements as well as full sets of energy-consistent Lagrangian shell equations are derived to determine the initial non-linear fundamental equilibrium path, the critical points of snap-through or bifurcation buckling and also the post-buckling deformations. The equations including all boundary conditions can be obtained in a compact form by using an appropriate operator description, which allows also to prove some properties as symmetry, positivity or positive definiteness, important for the construction of finite approximation procedures. The example of a shallow cylindrical shell panel with two opposite edges hinged and two opposite edges free under the action of a transverse point force and axial compression is calculated numerically. Using a simple Rayleigh-Ritz approximation it is shown that the collection of critical snap-through buckling points corresponds to the so-called cusp catastrophe, which is the second on a list of seven elementary catastrophes presented by Thorn [15].
t Paper presented at the EUROMECH 17-20 May 1983.
Colloquium 165 ‘Flexible Shells, Theory and Applications’, Mtinchen,
195
196
H. STUMPF 2. BASIC
SHELL
RELATIONS
The considerations of this paper are based on the physical assumptions that the shell is elastic, homogeneous and isotropic and ofconstant thickness h with (h/R) c-c 1, where R is the smallest principal radius of curvature of the undeformed shell middle surface. All shell deformations are assumed to be such that (h/L)’ << 1, where L is the smallest wave length of deformation pattern of the shell middle surface. Furthermore we presume that the strains are small everywhere in the shell, while the strain-displacement relations are non-linear. To obtain a Lagrangian description of the non-linear shell theory all shell deformations will be referred to the undeformed shell middle surface M. Let r(P), a = 1,2 be the position vector from a fixed Cartesian coordinate system to an arbitrary point of the reference shell middle surface as a function of a set of curvilinear Gaussian coordinates o”, a = 1,2. With 0” we associate covariant base vectors a, = r., and a unit normal vector n. The covariant components ofthe surface metric tensor aa and ofthe surface curvature tensor b,, are given by the scalar products Q, = ao.-a0 and b,, = a,,@‘II, where subscripts preceded by a comma indicate partial differentiation with respect to the corresponding coordinate direction. Using Einstein’s summation convention the contravariant components a”” of the metric tensor are defined by the relations a”“~A8= b”,, where 6; is the Kronecker symbol. Here and in all following formulas Greek indices take the values 1,2. The deformation of the shell middle surface from the undeformed reference configuration M into the deformed configuration R can be described by the displacement field u = ? - r = uaaa + u3n,
(2.1)
where u’ are the contravariant displacement components tangent to the middle surface and u3 is the displacement component normal to the middle surface. With the deformed shell middle surface l&iare associated corresponding covariant base vectors a, and a unit normal vector it defining the surface metric tensor ii,, and the surface curvature tensor 6,, of the deformed shell middle surface. Assuming that the state of stress is approximately plane and parallel to the middle surface or equivalently that the Kirchhoff-Love hypothesis is valid, the shell deformation can be described by the middle surface strain tensor yas and the change of curvature tensor ~~~
Yap =
;@a, -
a.,~),‘G/J =
-
(L? - bad.
(2.2)
Let t& be the linearized middle surface strain tensor and cpi, i = 1,2,3 the linearized rotations defined by t&l = f(Q?
+ Qln) -
b&3
cp. = UJ,, + b:u, with the contravariant
surface permutation
“,8
= ;@,,,
1 (P3 =
-
Qd
= $J(P3
(2.3)
jEaoU8,a
tensor
and a = det(a,,). In the frame of a non-linear shell theory with moderate rotations the strain measures (2.2) can be approximated by [S]:
Bucklq
and post-buckling
analysis of thin elastic shells
197
where a vertical stroke preceding a subscript indicates covariant differentiation with respect to the corresponding coordinate direction. With the modified tensor of elastic moduli
H =Pb
Eh
-
a=“aar + a=lJaal + -
-2(1
2v
l-v
a=P&
(2.5)
the stress resultant tensor NuP and the stress couple tensor Ma0 are given by the constitutive relations: N=D = H=aACylr = +
[(l - vhJ=p+ va=Py:]
(2.6) M=fl
=
!!tH=hK 12
Ap =
Eh3
12(1 - v2)
[(I -
V)K=’
+
VLf8K;],
where E is Young’s modulus and v Poisson’s ratio. Assuming that distributed surface loads per unit area of the undeformed middle surface are given by: P=
P% + w
(2.7)
the non-linear equilibrium equations for the moderate rotation shell theory are obtained in the following form [5]
(MuPI, +
1
+ f(OaaNf - BPaN;) ia - b;(MApjl + qANiB) + p= = 0
(2.8)
(p=NaP)I, + b=,[NaP - b”,M”P - w=“Nf] + p3 = 0.
Expressing N”” and MaB by the strain measures ?=aand K,+ and introducing then the straindisplacement relations (2.3)-(2.4) the equilibrium equations (2.8) yield a set of three nonlinear partial differential equations in thedisplacement components u,, a3 ofthe shell middle surface. To complete the set of governing equations by the boundary conditions we introduce an orthonormal vector system u, t, n at the line C bounding the undeformed shell middle surface M: dr f=z
= Pa,; u = t x n = v’a, = +PaP;n
1 2
= -caBam x a#,
(2.9)
where s is the length parameter of C. A cross indicates the usual vector product. With the physical components u, = u=v=,u, = U’C,and u3 the displacement vector u at the boundary can be expressed by u = u, 0 + u,t + u3n.
(2.10)
As a fourth independent geometric quantity at the boundary the rotation about the tangent to the boundary line By = -cp=d will be used. To describe the static boundary conditions the resultant stress vector T, and the resultant stress couple vector M, at the boundary are introduced, defined by T, = TBvb = Ty,.u+ TYt + T,,Vn M, = MPva = - M,,u+
M,,,t
(2.11)
198
H. STUMPF
with :
(3.12)
1
+ k(P”Nf: - OaaN;)
a, + (Mu% + (p.N”P)n
MP = &zlM”pa’. In (2.1 1) T,,,,, 7;,,, T,, and M,, = !@t,~~, M,,,. = MZPv,vp are physical quantities of the resultant stress and stress couple vector at the boundary. Let us assume that on a part C, of the boundary line C geometrical quantities and on a part C, statical quantities are prescribed. If geometrical and statical quantities are given on the same part of the boundary line, they must be complementary to each other. Then we have the geometrical boundary conditions on C, U, =
u:;u,
=
u:;u3
=
u$;fi,, = /?F;u~~ = uz,,j= l,..., N,
(2.13)
and the statical boundary conditions on C, [5]: v,vp [N”” - b;M”” - uY”Nf1] - b,,M,,, = T,:. - h,,M,f. = P;
- f(b;MaD
+ ;((P”Nf
+ b$M”“) - ;(b;M”fl
- @‘N:)
- bfMA”) - ;u’pN;
_ ;(w’“Ng
+ &.N;,)
1
- b,,M,, = T;c - b,,M;1. = P:
(2.14)
vp(MZ% + cp.N”“) + $ M,, = T,*, + $ M:, = P; v,v,, MmP= M $ Mlw(s/j + 0) - M,,(s,j
- 0) = M;C(s/j + 0) - M;F(s,
- 0) = Fr,j
= 1,. . . , IV,,
where b,, = v”tPbap is the geodesic torsion and b,, = t”tsb,, the normal curvature of C. Given values are indicated by an asterisk. If on the part C, discrete corner points are located at s =S,j,j=l,..., N, and if on the part C, discrete corner points are located at s = .s,~.j = 1, . . . , N/, then (2.13)5 and (2.14), are the geometrical and statical corner conditions. For all further considerations it will be presumed that external loads and boundary forces are of dead-load type. The equilibrium equations (2.8)expressed as functions ofthe middle surface displacement field II form together with the geometric boundary conditions (2.13) and the static boundary conditions (2.14) the full set of non-linear partial differential equations describing the nonlinear boundary value problem of elastic shells with moderate rotations. The solution of this boundary value problem yields the displacement field u depending non-linearly on the given external loads. 3. APPROPRIATE
OPERATOR
FORM
OF THE
LINEAR
SHELL
THEORY
In order to derive the equations defining the critical points of bifurcation or snap-through buckling and to determine the post-buckling deformations of shell structures in a unified and compact form we introduce an appropriate operator description of the shell theory. In this section we start with the linear shell theory and define linear shell operators with associated bilinear functionals which allow to investigate important properties of the shell operator as symmetry, positivity or positive definiteness. They are of special interest for the construction of numerical approximation procedures like finite element methods. From the relations of section 2 the linear shell equations are obtained by neglecting all non-linear terms. Assuming that the displacements u E H, are elements of a Hilbert-space H,
Buckling and post-buckling analysis of thin elastic shells
199
and that the middle surface forces PE H, ar; elements ofa Hiibert-space H,, we can put them into duality by a bilinear form (u, P) = j-l
M
(w’
(3.1)
+ U&dS
representing the work of the middle surface forces p applied to the displacement field u. Furthermore we define elements CEH, representing the state of strains and BE H, representing the state of stresses of the shell: II
\
NI
NZ2 Ni' &=
’
Ml1
(3.2)
M22
K22 K12
EHe.
+
M’2,
h’zl
The spaces H, and H, can be put into duality by the bilinear form:
<&VU) =
(O,,N;B +u,,M”P)dS JJM
(3.3)
with the linear stress resultant tensor: N;fi
= H”P”‘(j
(3.4)
AlI-
The bilinear form (3.3) is the interaction energy of two linear shell deformations described by E and CJ. We define now an operator F mapping elements ucDI c H, into H, by
J”
=
(3.5)
9
where in the subset DI t H, sufficient differentiability conditions have to be satisfied and rigid body motions are excluded. The constitutive equations of the linear shell theory can be expressed by using an operator
(3.6)
where N;#(O,,) and M@(K~,,) are given by (3.4) and (2.6),: Furthermore an operator .9*: H, + H, is introduced by
y*a=
-
N;fl - -I-(&M”fl + b@M”“) - ;(b;M”fl A 2 Ma”],, + bzp(Nra - b”,M”“)
- bfM”=)
IIP
b;MAP
200
H. STUMPF
which will be used to formulate the linear shell equilibrium equations in operator form. It is shown in equation (3.16) that the operator .Y* is formally adjoint to the operator .j/; The given definitions of linear shell operators and bilinear forms are represented graphically in the following scheme (Fig. 1):
Fig. I
According to Fig. 1 we can define an operator d with the domain D, = Dr and with the range H, by
Nf5(u) - ;(&M"(u) -
+ bf(M”“(u))-
;(&M”“(U)
b$M*“(u)q5 - bJM”BW12 EHP
(3.8)
MoP(~fl,p + b,s(NfP(u) - b”lh4@(u)) which allows one to formulate the linear equilibrium equations of the shell theory in the operator form : &u=p.
(3.9)
In a next step we have to introduce appropriate boundary operators. In the frame of the considered linear and non-linear, respectively, shell theory we define a geometric boundary vector ub representing the three physical components of the displacement vector u and the rotation /IVaround the boundary line C together with the normal displacement ujj at the discrete corner points i = 1, . . . . Correspondingly we define a static boundary vector P, collecting the three components P,, e, P3,the boundary stress couple M,,. and the corner forces Fj at the discrete corner points: PV f: ub
=
,Pb=
U3
PY U3j,j
=
l,...
p3 MVV Fj,j = l,...
1.
(3.10)
We put these geometric and static boundary vectors into duality by the line integral:
[ub,
Pbl
=
,-c @VP,+ u,p,+
u3P3 + PvMvv)ds + 5 u3jF,.
With (3.10) we define the following boundary operators:
j=l
(3.11)
201
Buckling and post-buckling analysis or thin elastic shells
V,UQ
1 I9 t,u=
nu =
UJ -
v3(u3..+ eh)
U,j,j =
1,. . . c
wpw~D-
b”lM”p) - b,,M,,
t=vg Ntp - ;(b”lMAB + b$M”=) - ;(b;M”@ [
‘== <
1
- b,,M,,
- trfM”=)
P. (3.12)
vsMaB(= + $M,.
v=vsMao ,Mtv(S/J + 0) - M,v(s/j - 0), j = 1,. . . , NJ
Corresponding to the operator & of the linear shell equilibrium equations we obtain a boundary operator ti ~=v~(N:~(u) - b$MAP(uj) - b,,M,Ju)
‘=vp Nfa(u) - ;(b”,M”B(u) + b$M”=(u)) - ;(b;M”‘(u) [ au = pZ3-u
=
QM=~I=W +$M,,(u)
- tiM”=(u))
1
- &MA) (3.13)
~=vpMuP(u) Mtv(u)(s/j + 0) - M,v(U)(S/j - O),j = 1,. . . , A’,
which will allow presentation of the static boundary conditions in operator form. The bilinear form (3.3) defines an energy product for linear shell deformations. For a strain field E(U)and a stress field a(v) (3.3) yields the interaction energy of the two shell deformations u and v: (s(u), a(v)) = fJ
mE~=@)NPB(v)+ K=&)MaB(v)ldS ~=,(U)&,(V) + g K=p(Uh,(“)
=
(3.14)
W
1
which, under certain restrictions, allows us to define an energy norm
lll411*=
1
=8aa dS jlrnH [Oaa(u)OA,(u)+ ;K,&)K&I)
=(&~u,.Tu) > 0
for u # 0,
(3.15)
because the operator X’ is positive definite for well-chosen elastic coefficients of (2.5). With (3.1) and (3.14) it can be shown by partial integration that the operator Y*, given by (3.7) is formally adjoint to the operator Y according to (3.5): (Yu,a)
= (u,Y*c)
+ [u,,pa].
(3.16)
It can be proved also that the operator LZ?of the linear shell boundary value problem is symmetric in a formal sense:
202
H.
(d-u, v) = (u, dv)
STUMPF
[p2r~u,v,]
-
(3.17)
+ [p~Yv,u,].
With the operators, introduced in this section, we formulate the linear shell boundary value problem in the following compact form: in M
du=p
KU= u,*
(3.18)
on C,,
au = Pb+ on C, where p are the given middle surface forces, II: the given geometrical boundary datas on C, and Pt the given statical boundary datas on C,. As solution of the boundary value problem (3.18) the displacement field u of the shell middle surface has to be determined.
4.
NON-LINEAR
SHELL
BOUNDARY
VALUE
PROBLEM
IN OPERATOR
DESCRIPTION
For the non-linear elastic shell theory with moderate rotations the middle surface strains yap depend non-linearly on the displacement field u. For further considerations we split the stress resultants Naa in a linear part NfB and a non-linear part N”,B Nap = Nf6 + N”6 m
(4.1)
with N;6
= H”6”‘(j
.Q
(4.2)
With the non-linear part Nip of the stress resultant tensor we introduce a non-linear operator w(u) by defining:
N”.‘(u)-
v?(u) = -
I
+a6(u)N:(u)
-
f(w"(u)N((u)
+ d"(u)Nf(u))
+ $=“(u)N$(u) - Bs’(u)N:(u)
2
II P
-
(4.3 1
b;dW%)
Associated with the operator U(u) we introduce a non-linear boundary operator c(u) given by:
I
v,v6
[Nt6(u)
-
w""(u)N!(u)]
t,v, Nz6(u) - +oa6(u)N:(u) - +“(u)Nf(u)
+ d”(u)N;(u))
+ ;(@=“(u)Nf(u) - @*(u)N’,(u))]
c(u) = <
’
(4.4)
vscp,(u)N”%) 0 (0
J
With the linear operators of section 3 and the non-linear operators (4.3) and (4.4) we formulate the non-linear shell boundary value problem of section 2 in operator description:
203
Buckling and post-buckling analysis of thin elastic shells
du + U(u) = p
in M
KU
= uh*
on C,
au + c(u)
= Pb*
on C,,
(4.5)
where (4.9, are the non-linear equilibrium equations (2.8) expressed in displacements, (4.5), are the geometric boundary conditions (2.13) and (4.5), are the non-linear static boundary conditions (2.14) in displacements. To derive the governing equations of the buckling and post-buckling analysis in a unified and compact form we introduce a non-linear row operator 9’(u) defined by:
9(u) =
du + W(u) in M au + c(u)
(4.6)
au + c(u)
The given middle surface forces p and the given statical boundary contracted to a given row vector F
variables Pt are
(4.7)
Furthermore
we. introduce a bilinear form {u,p) by (&PI = (SP) + [u*rPJ.
(4.8)
It will be shown in the following section that with (4.6)-(4.8) the non-linear shell boundary value problem (4.5) can be given in a variational formulation of the type {.9(u) - F,Q) = 0, ViE A,. 5. VARIATIONAL
PRINCIPLE OF TOTAL POTENTIAL ENERGY LINEAR SHELL BOUNDARY VALUE PROBLEM
FOR THE NON-
As solution of the non-linear shell boundary value problem (4.5) displacement fields Uare obtained, which, as functions of a one-parametric load (4.7), form a non-linear equilibrium path, where points of bifurcation or limit load points can occur. At bifurcation points the fundamental equilibrium path is intersected by a bifurcated path which can be stable or unstable. At limit load points the deformation of the shell structure becomes unstable. In this section a generalized Taylor expansion of the total potential energy of the shell is considered, which leads to a variational formulation of the non-linear shell boundary value problem. Let i denote the displacement field from a configuration ii to an adjacent configuration ii + ti: u=li+Q,
(5.1)
where it is assumed that ii satisfies the geometric boundary conditions (4.5), and that i satisfies the homogeneous geometric boundary conditions a&, = 0 on C,. All quantities will be referred to the known undeformed shell middle surface, which allows to derive the shell equations in a Lagrangian description. For the linearized field variables (2.3) and (2.4)2 ofthe adjacent state we have by superposition:
Inserting (5.2),-(5.2), following form:
into (2.4), the middle surface strain tensor yea is obtained in the
H. STUMPF
204
where yas and )iaDare given by (2.4), with corresponding indications. The total potential energy%,(u) of the moderate rotation shell theory is described by the functional [ 11,121 (P*
%Ju) = J/M Cp[Y&u), k~(u)] - p .u)dS - J
'U +
M,*,,BvIds - j$l FTuaj
(5.4)
cJ
where tE[~,~(u), Q(U)] is the strain energy density per unit area of the undeformed shell middle surface, given by:
(5.5) The first integral of (5.4) is taken over the undeformed shell middle surface M, the second integral along the boundary line C,, and the third term is the contribution ofall corner points of c,. The energy increment A$,(U; i) from a fundamental state ii to an adjacent state ii + ii is a measure for the behaviour of the shell structure at U.It can be derived by using a generalized Taylor expansion of the functional (5.4) leading to the series: A,$,(ii;u) = f&ii + C) - f,(u) = fb(ii;ii)
1 + -fb’(_;u i?) + ;$_.(P;Q”) 2
(5.6) + $s”(U;i”,,
where yb, yi, $ji, 2; denote the first, second, third and fourth Gateaux differentials of the total potential energy /*. They can be expressed in the following form by using the differentials of the strain energy density (5.5): N,
G”‘(ii;ii) - p.ii]dS
&(ii;
i.12)=
II
J”(U;
f;(I& $) = II f;(ij;i*)
-
(P* ‘ti + Mt,,/?,.)ds -
C Ffti,j
j= 1
Cti)dS
M/“(ii ; i.iiXi)dS
(5.7 1
=
All higher order differentials disappear. To calculate the first differential /‘(ii;fi), we have to consider the expression
limG + tii) - Au) 1-O
t
= /‘(U;C)
(5.8)
and if this limit is linear in 8, it is called the first Gateaux differential of the strain energy density Au). Let fi and ti be two different displacement fields superimposed upon the fundamental displacement field ii. The second differential of (5.5) is then defined by: limJ’(ii 1-O
+ tii; ii) - fyi-l; ii) = f”(ii; t
iii).
(5.9)
Differentials of the third and fourth order have to be calculated analogously. Introducing the strain tensor y.&(u)and the change of curvature tensor Q(U) according to (2.4) into (5.5), the differentials of Au) can be determined:
Buckling
and post-buckling
analysis of thin elastic shells
205
1 =P"y&(ii;O) +M=4z=p f'(ii;iii) = 1 (5.10) f(ii; ii) = Ha”“” $j=,ay;,(C;ii) + ;
i?=,k,,
Ha@“’ y:,&i; ii)y;,(U; ii) + j&y;,,(Li, ii) + ; II
k=gZir
j”‘(ir; riiifi) = HaBAr[2y&(ii; ii)y;,,(ti, ii) + y&&i; ii)y$,(i, ii)] f”‘(ii; Uii)
= 3HQBL”y&(ti,ti)yl;,(ii, ii),
where the following differentials of the strain tensor y=#(u) are used:
With (5.7), (5.10) and (5.11) the Taylor expansion (5.6) of the energy increment A%,,(ii;u) in the neighbourhood of a fundamental state ii is given. The principle of total potential energy yields the following statement. For all geometrically admissible displacement fields u, satisfying the geometric boundary conditions (4.5), the total potential energy (5.4) attains its stationary value f;(u;e)
= 0
VikA,;
7tu =
(5.12)
I$
at the solution u = ii. With A,, we denote a Hilbert-space of sufficiently differentiable displacement fields satisfying homogeneous geometric boundary conditions on C,. To derive the Euler-Lagrange equations associated with the variational statement (5.12), we introduce (5.1O)r into (5.7), and transform fb(u;ii) by partial integration leading to the expression : %Ju;i,
= - JI{[
(Nap - ;(b;Mip
- t(w=“NE + cd”Nf)
+ b$M”=) - ;(b;M”O - b!M”=) - fw=BN:
+ ;(Q=‘Nf - fIBAN;)
1
II P
- b;(M”@J, + cpINAP) + p” ti,
+
[(MapI= + cp=Naa)lp + b=,(N=@ - b”,MLB- w=“N{) + p,]li,}dS
[v=va(NaB- b”,M”# - w=‘Nf) - b,,M,, - I’;]&
+ 11C +
[
t=va NaB - ;(b;Mdb (
- ;(w=“Nf: + d”N”,)
+
+ bfM”=) _ ;(b;M”@ + +Nf 2
- @‘N;)
vB(Mn61=+ cp=N=“)+ $ M,, - P:
+
2
+ 0) -
1
1
_ tiMi=) _ fw”N;
1
- b,, M,, - P*, tiI
ii3 + (M,, - M,*,)j$
Mt,(s/j - 0) - F’T]ti,,
ds (5.13)
H.
206
STUMPF
where sufficient differentiability of II is assumed. Introducing (5.13) into the stationarity condition (5.12) yields the non-linear equilibrium equations (2.8) and the non-linear static boundary conditions (2.14) as Euler-Lagrange equations. To derive the Euler-Lagrange equations in the form (4.5) the membrane stress tensor N”” is split into the linear part NTBand the non-linear part N$’ with (4.1) and (4.2). Using the bilinear forms and operators of sections 3 and 4, the first differential of the total potential energy (5.13) can be expressed by: I;(u;ii)
= (SAI + 59(u) - p,C) +
[au+ C(U),&,]” + [cru+ c(u) - Pb*,h], (5.14)
= {Y(u) - F,4j. With (5.14) the stationarity
condition (5.12) is obtained in the compact form:
{.9(u) - F,ir} = 0
vii&,;
7cu
‘d(u) = p
in M
CLU+ c(u) = Pb*
on %,
=
u#f
(5.15)
(5.16)
defining the non-linear elastic shell boundary value problem. In the following sections it will be shown that the equations ofcritical equilibrium, defining snap-through and bifurcation buckling, and also the postbuckling equations can be given in a compact form by using the differentials of the operators d and .d, %%,(I, C, respectively. calculated analogue to (5.8) and (5.9). Because of lack of space they cannot be presented here but are outlined in detail in [16].
6. APPROACH
BY A SEQUENCE
OF LINEAR
SHELL
BOUNDARY
VALUE
PROBLEMS
The total potential energy f&F) is a functional of the displacement field u and of the external load F according to (4.7), where F may depend on various load parameters ij, j = 1, . . . , m. For many stability problems of engineering interest we can restrict our considerations to the study ofshell deformations along a fundamental equilibrium path, described by a oneparameter dependency ~j =
j
nj(~),
=
l,...,m,
(6.1)
where the external load F depends linearly on the load parameter i. F = if
(6.2),
with f denoting a unit load system:
(6.2),
Using (5.12) and (5.15) the stationarity condition for the fundamental path can be given in the two alternative forms: &(ii(A);i)
= 0
{P(ii@)) - Af,iif = 0
ViEt-9”
Vi&,.
(6.3)
We now consider the problem of solving (6.3) in the neighbourhood of a solution point 0 with known displacement field u0 by using the static perturbation technique of Sewell [ 171.
Buckling
and post-buckling
207
analysis of thin elastic shells
Introducing a parameter t with t = 0 at 0, it is assumed that the solution of (6.3) can be expressed by the parametric expansion ii(t) =
Ilo
+
ii(t)
= uo + tlqJ +
t3
t2 -u6’
+ -Ui,”
+
6
2
O(t4)
(6.4)
where a dot indicates differentiation with respect to the parameter t. An expansion of the same type is used to represent the load parameter A(t):
t3
t2 i.(t) = R, + rib + Tib.
+ 6ibb.. + O(t4).
(6.5)
With (6.4) the governing shell operator :f(u(A)) according to (4.6) can be expanded in a Taylor series: .9(ii(t))
=
squ, + i(t))
= !Y(Uo) + tY(u,)u~
+ f
+ ~Y’(uo)ub’]
[w(u&l~
+ ; [9’(U~)U&. + 3.9”(U,)U~Ub.+ ~“‘(UO)U~] + O(t4). (6.6 1 Introducing (6.5) and (6.6) into the stationarity condition (6.3), the following sequential variational statements are obtained: (d(uo) - A”f,iif = 0 - &f, i) = 0 {8’(U&~. + d”(U,)U& - &f,ti) = 0 {W(uo)ub.. + 3Y’(u,)ubub. + d”‘(u,)ub3 - j.b.*f,d) = 0 { d’(u,)ug
vi Vii vi Vi (6.7)
The sequential stationarity conditions (6.7) define a sequence of shell boundary value problems, which allow to determine the unknown coefficients of expansion (6.4). The Euler-Lagrange equations corresponding to (6.7), are the non-linear equilibrium equations and the non-linear static boundary conditions duo + %(uo) = lop.,
au0 + c(uo) = loPb,
in M on C,
(6.8)
with the solution uo. We assume that the solution point 0 is a regular point with a unique solution u. of (6.8). Then (6.7), yields the linear equations: .G/u~ + ~‘(u,)u~ = Ibp, &lb + C’(Uo)U~= &P,*
in M on C,
(6.9)
depending non-linearly on uo. The solution of this linear boundary value problem is the displacement field ug . The variational statement (6.7), leads to the equations .&I~.
+
V’(U,)U~
UUb’+ c’(u&j
=
&p,
-
V”(uo)u&
= &P,* - c”(u,)ui:
in M on C,
(6.10)
208
H. STUMPF
with the solution uv. Correspondingly the variational statements (6.7),, (6.7!,, . . . define linear shell boundary value problems with the solutions ug., ub’.., . . . , the coefficients of expansion (6.4). If this approximation procedure is stopped after n terms the equilibrium error can be calculated, this will not be considered here because of lack of space. 7. CRlTlCAL
POINTS
OF THE
FUNDAMENTAL
PATH
The determination of the critical points of the fundamental path can be performed by considering the behaviour of the total potential energy yJu(t), L(t)) in the vicinity of a solution point uM, &, along every possible one-parameter deformation a(t) u(t) =
u&f +
i(t)
CU’ +
=
i(t)
fu.. +$... +0(t4)
(7.1)
2 = A(t) = AM
superimposed upon the deformation uM, where theload parameter1retains the constant value AM* The energy increment (5.6) at the solution point uy can be obtained as Taylor expansion : A~&,;ii(t))
= ;&‘(u,;u’)
+ f [3~5(u~;uu~.)
+ $&‘(uI(;u~~~.) + fF”(U&&+)]
+ ~~;(u~;u..~)
+ &“(u,;u3)] + ~&‘(u~;u.~u.~)
+ O(?).
(7.2)
It will be shown that at critical points the first term on the RHS of (7.2) will vanish. We have to investigate whether the one-parameter deformation (7.1) can be also an equilibrium deformation, such that the sequential stationarity conditions (6.7) are satisfied for constant I-values. In this case all derivatives of 1 vanish and (6.7) yields:
{!9yuhl)- &#f,ii} = 0 {Y(u,)u.,ii) = 0 (9’(U~)U.. + 9”(U&v2, ii) = 0 {~‘(Uy)U~~~+ 3Y’(uy)uu.. + 8”‘(Uhl)U~3,P) = 0
vii vi vi
(7.3)
vi
Thevariational statement (7.3)r is the stationarity condition for the solution point uM,i., , which is also the trivial solution of (7.3),, (7.3),, . . . for II*= II*+= . . . = 0. If we compare (7.3), with (6.7), it is obvious that (7.3)* has no solution in the non-critical case and therefore also (7.3),, . , . cannot be satisfied. Only in the critical case, denoted by AM = AC,;uhf
and satisfying the stationarity
=
(7.4),
II,
condition {P(uc) - A,f, 6} = 0
vii
(7.4)*
there may be one or more solutions of the variational statement:
{B’(U,)ll~,Q}= 0 Vii
(7.5)
with the restriction that II; has to satisfy some normalizing condition. The index c refers to the critical
point.
Buckling and post-buckling
209
analysis of thin elastic shells
The solution equations of (7.5) yield the conditions for critical equilibrium, the equations of snap-through or bifurcation buckling of shells: du; + ~‘(u~)u; = 0 au; + c’(u,)u; = 0
in M on C,
(7.6)
with the linear operators & and a, defined by (3.8) and (3.13). The operators U’(u,) and c’(u,) are the Ghteaux derivatives of (4.3) and (4.4) and depend non-linearly on the prebuckling deformation u,. The linear equation (7.6), represents three homogeneous field equations of the shell middle surface and (7.6), the associated static boundary conditions. They define a homogeneous eigenvalue problem depending non-linearly on u,(&). The solution of the eigenvalue problem (7.5) and (7.6), respectively, yields the buckling mode u; with the associated buckling load I,. To normalize the buckling mode u; we choose a normalization according to the strain energy of linear shell deformations by setting: {Y’(o)u;,u~> = 1.
(7.7)
Here Y(0) denotes the derivative of the operator P(u) at u = 0. Using definition (4.6) of the operator Y(u) and definition (4.8) of the bilinear form {. , .}, expression (7.7) represents the double elastic shell energy for the small deformation u;: {d’(O)u;,u~) = (du;,u;)
+
[au;,IQ,] O,&)OA,(u;) + ; K~,#;)K&)
1
dS.
(7.8)
Using u; to define shell buckling modes, higher order contributions ofexpansion (7.1), are not considered and the question, whether or not (7.3),, . . . are satisfied, is not discussed. If we choose in (7.5) ii = U;E& we can prove easily: {Y’(u,)ll;,
u;}= y;(u,;u;2) = 0,
(7.9)
leading to the result that the first term of the energy expansion (7.2) vanishes at uc, I,. This corresponds to the energy criterion of stability that for critical deformations the second differential of the total potential energy must disappear. If we compare the stability equations (7.6) with the governing equations (4.5), and (4.5), of the non-linear shell boundary value problem, it is obvious that (7.6) is the differential of (4.5), and (4.5), for A(t) = A,. This gives a very general form of the so-called adjacent equilibrium criterion well-known in the engineering literature [ 181. 8. BIFURCATION
AND
POST-BUCKLING
ANALYSIS
Let us assume that, at 3, = AC,, a new equilibrium path ub(A)bifurcates from the fundamental path ii(A): l+(A) = ii(A) + v(A).
(8.1)
With v(A)we denote the differential displacement field from the fundamental to the bifurcated path. Under suitable regularity conditions the fundamental equilibrium path ii(A) can be expressed by the Taylor expansion near the bifurcation point u,, A,: ii(l)
=
UC +
(A -
;(A- It,,‘lz””
n,)uup+
+
;(A- ~c)3~ooo+ W
- A,r),
(8.2)
where the symbol ( )’ indicates differentiation with respect to the load parameter I, ( )’ d( ) The index c refers to the bifurcation point. = =. L
210
H. STUMPF
For the differential displacement v(A(t)) we use the parametric representation at the bifurcation point: t2 t3 v(t) = tv; + TV;.’ + gv;.
The corresponding
. +0(P).
with (t = 0)
(8.3)
expansion of the load parameter l(t) near t = 0 yields: l(t)
= 2, + tl; + GA;.
+ :A;.+
+ O(t”).
(8.4)
The coefficients of the parameter expansion (8.3) may be normalized according to the bilinear form (+“(O). , .) : {Y(O)v;,v;}
= 1
(8.5)
{Y’(O)v;, vi.1 = {9’(O)v;, Vi..} = . . . = 0.
Assuming that the bifurcated path u”(i) is an equilibrium path the stationarity condition (~5.3)~must be satisfied: (@)
- lf,ii} = 0
vii
(8.6)
With (8.1) we use the Taylor expansion of the operator Y@“(i)): .?P(Uh(;L))= 9(U) + Y’(ti)v
yielding the two stationarity
+
;~t’(u)v2 ++(u)v’
(8.7)
conditions: VI-i
(9yi-i) - J&f,ii} = 0
(8.8 { P’(U)v + ;Y(ii)v2
+ ++‘“(ti)v’, ti) = 0
vi
Relation (8.8), is the stationarity condition for the fundamental path and (8.8), the stationarity condition to define the bifurcated path. For further considerations it is presupposed that the solution of (8.8),, the fundamental path (8.2), is known. The operators of (8,8)2,9’(U), Y’(U) and Y”(ii) will be expanded at the bifurcation point. With (8.2) and (8.4) we obtain the series: 9’(ii(i(t)))
= W(u,) + tiLy’(U,)up+ +
[~~2w’(u,)upo+ i~;GP”(u,)up
;
~;29”‘(u,)up2] + o(t3)
P”(ii(A(t)))
= P"(u,)
+ ti.~"'(u,)u~+
d”‘(ii(A(t)))
= W”(U,) = 9”‘.
(8.9)
O(t2)
We introduce the operators (8.9) into the stationary condition (8.8)2. Replacing v by (8.3) and collecting the various terms with respect to the powers oft, the following result can be derived : Vi
{Y(u,)v;, ii} = 0 {Y’(uc)v;. + P”(u,)v;2 + 2~;~“(u,)upv~,o)
v’e
= 0
{ Y’(u,)v;~ . + 3~“(u,)v;v;~ + Y”(llJV;3 + 31;[~“(u,)upv;*
+ Y”‘(Ur)U;V;2]
+ 32;2 [.V’(uC)up”v~ + B”‘(u,)up2v;] + 31;~B”(u,)upv;, ti) = 0 vi
(8.10)
Buckling
and post-buckling
analysis of thin elastic shells
211
representing a sequence of stationarity conditions to determine the bifurcated path (8.1) The first variational statement (8.1O)r describes together with the normalization condition (8.5), the homogeneous eigenvalue problem of bifurcation buckling of thin shells. With definition (4.8) of the bilinear form {. , .} and with the operator 9’(u,) the solution equations of (8.10), dv; aV;
+ U’(u,)v; = 0 +
e’(U,)V;
=
0
in M
(8.11)
on C,
are the stability equations, depending non-linearly on the prebuckling deformation II,(&). Equation (8.1 I), represents the field equations and (8.1 1)1 the associated static boundary conditions. The solution of the eigenvalue problem (8.11) yields the buckling mode v; together with the bifurcation load parameter A,. In this section it is assumed that ACis a singular eigenvalue. Because of (8.1) the increment of the bifurcated deformation at the bifurcation point is: Ub. =
u; + v;.
(8.12)
To determine the coefficient rl; of series (8.4), we choose in (8.10)2 ii = v;E&. symmetry of the operator F(u,) [16] we obtain with (8.10),: {9’(ll$;~,
Vi} = {v;.,B’(u,)v;;
= 0
Using the
(8.13)
Then (8.10)2 yields for ii = v;: *. = c
_A
P”(~)v~2~ Vi)
2 {s”“(u,)tJpv;, v;}’
(8.14)
where it is assumed that {B”(u,)u~v;.,v;} # 0. With known v; and rl; (8.lO)z is a variational statement to determine the coefficient v;.. The associated solution equations J&Q;*+ %?‘(uc)v;*= - ~“(uc)v;2 - 2A;V’(u,)upv; (8.15) nv;. + G’(U,)V;’= -c”(llJV;~ - 2i;c”(t4)$%; form a set of linear nonhomogeneous partial differential equations, describing a linear shell boundary value problem yielding the displacement vi*. The first line in (8.15) represents the field equations, the second line the associated static boundary conditions. The solution of (8.15) is not unique, because any multiple of v, can be added to a particular solution of (8.15), which we denote by vi;. To satisfy the normalizing condition (8.5), the unique solution v;* is given by: Vi’
=
v;; - { B’(O)v;;,
v;}v;,
(8.16)
which can be shown by introducing (8.16) into (8.5),. Continuing the procedure we choose in (8.10)3 i = v;~fi, such that {~‘(u,)v;~.,v;~ disappears because of the symmetry of Y(u,) and because of (8.1O)r. Then the load coefficient 1;. can be determined from (8.10)3 yielding:
{a”(U,)V;V;.’ +
),; . =
-
+
f.s~~~(uJY;” +&[9”(u,)upv~* + Y”‘(u,)upv;2] Ai2[w’(uJup” v; + ~“‘(uc)llp2v;],Vif {3p”(u,)upv;,vi;
(8.17)
H. STUMPF
212
For many shell buckling problems the bifurcation is symmetric. In this case the coefficient 1; of the load parameter expansion (8.4) vanishes leading to a considerably simplified expression for I; .:
{.~“(u,)v;v;~+ A;. =
-
f9’.‘(uc)v;J. v;.)
{Y’(u,)upv;,v;}
.
(8.18)
With known A;, A;-, v;, v;. expression (8.lO)3 is a variational statement to determine the displacement field v; . .. The associated solution equations define a linear shell boundary value problem, which allows one to determine vi**.Analogously the further variational statements of (8.10) yield the higher order terms of the series (8.3) and (8.4) describing the bifurcated equilibrium path. With the load coefficient (8.14) of the parameter expansion (8.4) we obtain a necessary condition for stability of the shell at bifurcation points { P”(U,)Vf2,Vi} = 0
(8.19)
or, equivalently, 1; = 0. A further necessary condition for stability is {.GY”‘(u,)v;3, v;} - 3{8’(u,)v;~,v;~} > 0,
(8.20)
for which the increment of total potential energy Afp(uc;ub) is positive. The homogeneous eigenvalue problem (8.11) of shell buckling may have n linearly independent solutions, the eigenmodes v;(~),i = 1, . . . , n associated with a multiple buckling load d,. In this case we have to modify slightly the post-buckling analysis of this section by introducing a multiple buckling mode [16]. 9. NUMERICAL EXAMPLE Let us consider a circular cylindrical shallow shell panel with two opposite edges hinged and two opposite edges free (Fig. 2). Using a finite element procedure this example had been calculated in [19] for a central point force P and in [20] for a combined loading of central point force P and axial compression N under the assumption of a fixed ration N/P. This assumption leads to a one-parameter deformation represented by an equilibrium path. Here we want to investigate the more general case of a two-parameter dependency such that P and N can be varied independently. Furthermore an imperfection or eccentricity E of the point load P is introduced (Fig. 2): The geometry of the shell panel with constant thickness h, radius R, length 2L and opening angle cpo can be described by two shell parameters
Fig. 2. Circular
cylindrical
shell panel with eccentric load P and axial load N.
Buckling and post-buckling analysis of thin elastic shells
213
= R&/h and p2 = RqdL. With IA,v, w, we denote the axial, circumferential and normal displacement components in nondimensional form. The displacement field (a, v, w) has to satisfy the geometric boundary conditions at the hinged edges: v = w = 0. Using only nondimensional quantities the total potential energy of shells according to (5.4) leads for the shell problem of Fig. 2 to a functional f = f(u, v, w, pl, p2, P, N, E) depending on the displacement components u, v, w, two shell parameters cc,, p2, two load parameters P, N and the eccentricity E. To determine the stability boundaries of this shell structure a Rayleigh-Ritz approximation can be used by choosing a set of geometrically admissible displacement fields with unknown coefficients, complete in a well-defined space. For a first approximation we choose Pl
u(x) = -Usinix
v(q) = Vsin ncp
(9.1)
w(x,q) = W, cosicp + W,sin7rq
with the unknown coefficients U, V, W,, W,. Assumption (9.1) satisfies the geometric boundary conditions at x = x/L and cp = (p/q,. Introducing (9.1) into the total potential energy (5.4) leads to a functional %
=
f(u,
K
w,,
K
PI,
~2,
P,
N,
~1.
(9.2)
Differentiation with respect to the unknown coefficients yields four equilibrium equations: 8%
-
au =
0;: =o;g=o;g=
0.
s
(9.3)
Y
Equations (9.3), and (9.3), are linear in U and V, respectively. Eliminating U and V and introducing them into equations (9.3), and (9.3)., gives two non-linear equilibrium equations in the unknown coefficients W, and W,. The snap-through buckling of the cylindrical shell panel will be governed by symmetric deformations. Therefore we can choose the coefficient for unsymmetric deformation W, = 0 and obtain one equilibrium equation: W,3e3 + W,2e2 + W,e, + e. = 0
(9*4)1
with: eo=-
128v/n3
N --
2
p.,
e2 = - 8.737
Ccl
2.029” 2 4 e, = 3.043 + 2 -2-N; Pl
Using the transformation
(9.4)~ e3 = 5.644
Pl
W, = W - $
and putting 3
(9.5) yields an equilibrium surface defined by:
M = {(w, f0,fl)lW3 -f1W+fo
= 0).
(9.6)
214
H. STUMPF
The critical points of the equilibrium surface (9.6) are the snap-through buckling points, for which the second differential of the total potential energy vanishes according to (7.9) yielding:
Expression (9.7) represents the collection of critical snap-through buckling points for a twoparameter loading and corresponds to a cusp-catastrophe [15]. The geometrical representation is given in Fig. 3 showing the collection of upper and lower snap-through buckling loads as well as the areas of stable and unstable behaviour of the shell. Figure 4 shows the equilibrium surface (9.6) in the three-dimensional (W,, P, P/)-space and the collection of critical snap-through buckling loads in the (P, N)-plane. Furthermore we have to investigate the unsymmetric bifurcation buckling by taking into consideration unsymmetric deformations with W, # 0, E # 0. It can be shown that for special values of axial compression N, bifurcation buckling exists and the collection of critical bifurcation buckling loads can be determined analogue to the results for symmetric snapthrough buckling. CUSP - CATASTROPHE
lower
critical
Fig. 3. Cusp catastrophe
of snap-through
Equilibrium
/N
Fig. 4. Equilibrium
buckling for two-parameter
loading
Surface /
/
Collection ot Critical
surface and cusp catastrophe
Points or snap-through
buckling
({I, = 1)
Buckling and post-buckling analysis of thin elastic shells
215
Arkno*,le~gements-Theauthor would like to thank the Polish Academy ofSciences inGdadsk for partial support of this work and Prof. Dr. W. Pietraszkiewicz for stimulating discussions. REFERENCES 1. W. T. Koiter, On the stability of elastic equilibrium, Ph.D. Thesis, Polytechnic Institute. Dclft (1945). NASA (translation) TTF-IO, 833 (1967). 2. W. T. Koiter, Elastic stability and post-buckling behavior. In Proceedings of rhe Symposium on Non-linrur Problems, (Edited by R. E. Langer), pp. 257-275. University of Wisconsin Press (1963). 3. W. T. Koiter, General equations of elastic stability for thin shells.In Proceedings ojthe Symposium on Theory oj Shells to Honour Lloyd H. Donnell, pp. 187-227. University of Houston (1966). 4. B. Budiansky, Theory of buckling and post-buckling bchaviour of elastic structures, Adu. appl. Mech. 14, 1-65 (1974).
5. W. Pietraszkiewicx, Introduction to the non-linear theory of shells, Mirr. Inst. Mech. 10, Ruhr-Univ. Bochum (1977). 6. 1. L. Sanders, Non-linear theories or thin shells, Q. uppl. Math. 21, 21-36 (1963). 7. W. T. Koiter, On the non-linear theory of thin elastic shells, Proc. Kon. Ned. A&. Wet., Ser. B, 69, 1-54 (1966). 8. L. P. Nolte, Beitragzur Herleitung und vergleichende Untersuchunggeometrisch nichtlinearer Schalentheorien unter Berticksichtigung groBer Rotationen, Mitt. Inst. Mech. 39. Ruhr-Univ. Bochum (1983). 9. W. Pietraszkiewicz, On consistent approximations in the geometrrcally non-linear theory ol’shells, Mirf. In~r. Mech. 26, Ruhr-Univ. Bochum (1981). 10. L.-P. Nolte and H. Stumpf, Energy-consistent large rotation shell theories in Lagrangean description. Mech. Rex Comm. 10.213-221 (1983). Il. H. Stumpf, The derivation of dual extrcmum’and complementary stationary principles in geometrical nonlinear shell theory, Irrg. Arch. 48, 221-237 (1979). 12. R. Schmidt and W. PietraszkiewicG Variational principles in the geometrically non-linear theory of shells undergoing moderate rotations, Ing. Arch. 50, 187-201 (198 1). 13. H. Stumpf. On the linear and non-linear stability analysis in the theory of thin elastic shells, Lecture XV ICTAM Toronto/Canada 1980, Ing. Arch. 51, 195-213 (1981). 14. H. Stumpf, The stability equations ofthe consistent non-linear elastic shell theory with moderate-rotations. In I UTAM-Symposium: Srabiliry in the Mechanics oj Coarintta, Niimbrecht/Germany 1981, Proc. 89- 100. Springer, Berlin (1982). Srahilify und Morpliogcnesis. Benjamin, Reading, MA (1975). 15. R. Thorn, Swwrural 16. H. StumpL Undied operator description, non-linear buckling and post-buckling analysis ofthin elastic shells. Mitt. Inst. Mech. 34, Ruhr-Univ. Bochum (1982). 17. M. J. Sewell, A general theory of equilibrium paths through critical points, 1, II. Proc. R. Sot. A306,201-223, 225-238 (1968). 18. D. 0. Brush and B. 0. Almroth, Buckling of Bars, P/aces and Shells. McGraw-Hill, New Yqrk 1975. 19. A. B. Sabir and A. C. Lock, The application of finite elements to the large deflection geometrically non-linear behaviour of cylindrical shells. In Variationul Methods in Engineering, (Edited by C. A. Brebbia and H. Tottenham), 7/54-7/65. Southampton, University Press, Southampton (1972). 20. E. Stein, A. Berg and W. Wagner, Different levelsofnon-linear shell theory in finite element stability analysis, In Buckling CI/ Shells. (Edited by E. Ramm), Springer. Berlin (1982).
Zusarmrenfassung: In der vorliegenden Arbeit wird die nichtlineare Stabilitatsund Nachbeulberechnung elastischer Strukturen im Ftahmen einer geometrisch-nicht-linearen Schalentheorte mit moderaten Rotationen untersucht. Unter Benutrung einer vollstandig Lagrangeschen Betrachtungsweise werden Variationsaussagen sowie die rugehorlgen Systeme nichtlinearer Schalenglelchungen unter ElnschluB der Randbedingungen hergeleitet. die eine Berechnung des fundamentalen Gleichgewichtspfades, kritischer Punkte mit Durchschlagsoder Verzweigungsbeulen sowie der NachbeuldeUrn die erhaltenen Glelchungen formationen ermoglicben. In einer kompakten Form darstellen zu konnen, wird eine Dlese verelnfachende Operator-schreibweise eingefuhrt. ermoglicht daruber hinaus den Nachweis elniger msentlicher Eigenschaften, die zur Ronstruktion.von Approximationsverfahren wie Finite-Element Methoden von Bedeutung sind. Ein Beispiel wird unter Ritz Ansatzes numerlsch fur eine Zmi-Parameter Durchschlagspunkte eine
Benutrung eines einfachen Rayleighuntersucht. Es wird gezeigt, da8 Belastung die Henge der kritischen Ketastrophe vom Typ “Cusp” ist.
+ .W Press Ltd.
ON THE NON-LINEAR BUCKLING AND POSTBUCKLING ANALYSIS OF THIN ELASTIC SHELLS? H. Ruhr-Universitlt
STUMPF
Bochum, Lehrstuhl fur Mechanik 11, Universit6tsstraBe lSOlA, D-4630 Bochum I, Federal Republic of Germany (Receiued 3 June 1983)
Abstract-ln this paper the non-linear stability and post-buckling analysis of elastic structures is considered in the frame of a geometrically non-linear shell theory with moderate rotations. Using a total Lagrangian description variational statements as well as associated sets of shell equations including boundary conditions are derived to determine the fundamental quilibrium path, critical points of snap-through or bifurcation bucklingand also the post-buckling deformations. To present these equations in a compact form a unified operator description is introduced. It also allows one to prove some important properties, which are needed to construct appropriate approximation procedures like finite element methods. A shell example is calculated numerically by using a simple Rayleigh-Ritz approximation. It is shown that for a two-parameter loading the collection of critical snap-through buckling points is a catastrophe of the *cusp’ type.
I. INTRODUCTION
Basic contributions to the stability and post-buckling analysis of elastic shell structures had been given by Koiter in several papers, among which [l-3] are mentioned here. Using a general energy functional Budiansky investigated the theory of buckling and post-buckling ofelastic structures [4], where some applications are also given. In the literature the stability behaviour of many special shell structures has been the subject of a vast numer of further publications. The aim of this paper is to consider the buckling and post-buckling of elastic structures in the frame of a general non-linear elastic shell theory for which we use the energy-consistent so-called ‘moderate rotation shell theory’, characterized by the assumption of small strains and moderate rotations [5]. This theory leads to an adequate description of a wide area of shell problems of engineering interest and contains many simplified theories as those connected with the names of Sanders [6], Koiter [7] and Donnell-Marguerre-Vlasov. It is shown in [8] that only for one-dimensional problems as beams and columns and some very special shell structures in the post-buckling range a ‘large rotation shell theory’ [9, lo] can lead to remarkable different results. For the moderate rotation shell theory variational principles have been derived in [ 11, 123, while the Lagrangian stability equations are given in [13,14]. In this paper variational statements as well as full sets of energy-consistent Lagrangian shell equations are derived to determine the initial non-linear fundamental equilibrium path, the critical points of snap-through or bifurcation buckling and also the post-buckling deformations. The equations including all boundary conditions can be obtained in a compact form by using an appropriate operator description, which allows also to prove some properties as symmetry, positivity or positive definiteness, important for the construction of finite approximation procedures. The example of a shallow cylindrical shell panel with two opposite edges hinged and two opposite edges free under the action of a transverse point force and axial compression is calculated numerically. Using a simple Rayleigh-Ritz approximation it is shown that the collection of critical snap-through buckling points corresponds to the so-called cusp catastrophe, which is the second on a list of seven elementary catastrophes presented by Thorn [15].
t Paper presented at the EUROMECH 17-20 May 1983.
Colloquium 165 ‘Flexible Shells, Theory and Applications’, Mtinchen,
195
196
H. STUMPF 2. BASIC
SHELL
RELATIONS
The considerations of this paper are based on the physical assumptions that the shell is elastic, homogeneous and isotropic and ofconstant thickness h with (h/R) c-c 1, where R is the smallest principal radius of curvature of the undeformed shell middle surface. All shell deformations are assumed to be such that (h/L)’ << 1, where L is the smallest wave length of deformation pattern of the shell middle surface. Furthermore we presume that the strains are small everywhere in the shell, while the strain-displacement relations are non-linear. To obtain a Lagrangian description of the non-linear shell theory all shell deformations will be referred to the undeformed shell middle surface M. Let r(P), a = 1,2 be the position vector from a fixed Cartesian coordinate system to an arbitrary point of the reference shell middle surface as a function of a set of curvilinear Gaussian coordinates o”, a = 1,2. With 0” we associate covariant base vectors a, = r., and a unit normal vector n. The covariant components ofthe surface metric tensor aa and ofthe surface curvature tensor b,, are given by the scalar products Q, = ao.-a0 and b,, = a,,@‘II, where subscripts preceded by a comma indicate partial differentiation with respect to the corresponding coordinate direction. Using Einstein’s summation convention the contravariant components a”” of the metric tensor are defined by the relations a”“~A8= b”,, where 6; is the Kronecker symbol. Here and in all following formulas Greek indices take the values 1,2. The deformation of the shell middle surface from the undeformed reference configuration M into the deformed configuration R can be described by the displacement field u = ? - r = uaaa + u3n,
(2.1)
where u’ are the contravariant displacement components tangent to the middle surface and u3 is the displacement component normal to the middle surface. With the deformed shell middle surface l&iare associated corresponding covariant base vectors a, and a unit normal vector it defining the surface metric tensor ii,, and the surface curvature tensor 6,, of the deformed shell middle surface. Assuming that the state of stress is approximately plane and parallel to the middle surface or equivalently that the Kirchhoff-Love hypothesis is valid, the shell deformation can be described by the middle surface strain tensor yas and the change of curvature tensor ~~~
Yap =
;@a, -
a.,~),‘G/J =
-
(L? - bad.
(2.2)
Let t& be the linearized middle surface strain tensor and cpi, i = 1,2,3 the linearized rotations defined by t&l = f(Q?
+ Qln) -
b&3
cp. = UJ,, + b:u, with the contravariant
surface permutation
“,8
= ;@,,,
1 (P3 =
-
Qd
= $J(P3
(2.3)
jEaoU8,a
tensor
and a = det(a,,). In the frame of a non-linear shell theory with moderate rotations the strain measures (2.2) can be approximated by [S]:
Bucklq
and post-buckling
analysis of thin elastic shells
197
where a vertical stroke preceding a subscript indicates covariant differentiation with respect to the corresponding coordinate direction. With the modified tensor of elastic moduli
H =Pb
Eh
-
a=“aar + a=lJaal + -
-2(1
2v
l-v
a=P&
(2.5)
the stress resultant tensor NuP and the stress couple tensor Ma0 are given by the constitutive relations: N=D = H=aACylr = +
[(l - vhJ=p+ va=Py:]
(2.6) M=fl
=
!!tH=hK 12
Ap =
Eh3
12(1 - v2)
[(I -
V)K=’
+
VLf8K;],
where E is Young’s modulus and v Poisson’s ratio. Assuming that distributed surface loads per unit area of the undeformed middle surface are given by: P=
P% + w
(2.7)
the non-linear equilibrium equations for the moderate rotation shell theory are obtained in the following form [5]
(MuPI, +
1
+ f(OaaNf - BPaN;) ia - b;(MApjl + qANiB) + p= = 0
(2.8)
(p=NaP)I, + b=,[NaP - b”,M”P - w=“Nf] + p3 = 0.
Expressing N”” and MaB by the strain measures ?=aand K,+ and introducing then the straindisplacement relations (2.3)-(2.4) the equilibrium equations (2.8) yield a set of three nonlinear partial differential equations in thedisplacement components u,, a3 ofthe shell middle surface. To complete the set of governing equations by the boundary conditions we introduce an orthonormal vector system u, t, n at the line C bounding the undeformed shell middle surface M: dr f=z
= Pa,; u = t x n = v’a, = +PaP;n
1 2
= -caBam x a#,
(2.9)
where s is the length parameter of C. A cross indicates the usual vector product. With the physical components u, = u=v=,u, = U’C,and u3 the displacement vector u at the boundary can be expressed by u = u, 0 + u,t + u3n.
(2.10)
As a fourth independent geometric quantity at the boundary the rotation about the tangent to the boundary line By = -cp=d will be used. To describe the static boundary conditions the resultant stress vector T, and the resultant stress couple vector M, at the boundary are introduced, defined by T, = TBvb = Ty,.u+ TYt + T,,Vn M, = MPva = - M,,u+
M,,,t
(2.11)
198
H. STUMPF
with :
(3.12)
1
+ k(P”Nf: - OaaN;)
a, + (Mu% + (p.N”P)n
MP = &zlM”pa’. In (2.1 1) T,,,,, 7;,,, T,, and M,, = !@t,~~, M,,,. = MZPv,vp are physical quantities of the resultant stress and stress couple vector at the boundary. Let us assume that on a part C, of the boundary line C geometrical quantities and on a part C, statical quantities are prescribed. If geometrical and statical quantities are given on the same part of the boundary line, they must be complementary to each other. Then we have the geometrical boundary conditions on C, U, =
u:;u,
=
u:;u3
=
u$;fi,, = /?F;u~~ = uz,,j= l,..., N,
(2.13)
and the statical boundary conditions on C, [5]: v,vp [N”” - b;M”” - uY”Nf1] - b,,M,,, = T,:. - h,,M,f. = P;
- f(b;MaD
+ ;((P”Nf
+ b$M”“) - ;(b;M”fl
- @‘N:)
- bfMA”) - ;u’pN;
_ ;(w’“Ng
+ &.N;,)
1
- b,,M,, = T;c - b,,M;1. = P:
(2.14)
vp(MZ% + cp.N”“) + $ M,, = T,*, + $ M:, = P; v,v,, MmP= M $ Mlw(s/j + 0) - M,,(s,j
- 0) = M;C(s/j + 0) - M;F(s,
- 0) = Fr,j
= 1,. . . , IV,,
where b,, = v”tPbap is the geodesic torsion and b,, = t”tsb,, the normal curvature of C. Given values are indicated by an asterisk. If on the part C, discrete corner points are located at s =S,j,j=l,..., N, and if on the part C, discrete corner points are located at s = .s,~.j = 1, . . . , N/, then (2.13)5 and (2.14), are the geometrical and statical corner conditions. For all further considerations it will be presumed that external loads and boundary forces are of dead-load type. The equilibrium equations (2.8)expressed as functions ofthe middle surface displacement field II form together with the geometric boundary conditions (2.13) and the static boundary conditions (2.14) the full set of non-linear partial differential equations describing the nonlinear boundary value problem of elastic shells with moderate rotations. The solution of this boundary value problem yields the displacement field u depending non-linearly on the given external loads. 3. APPROPRIATE
OPERATOR
FORM
OF THE
LINEAR
SHELL
THEORY
In order to derive the equations defining the critical points of bifurcation or snap-through buckling and to determine the post-buckling deformations of shell structures in a unified and compact form we introduce an appropriate operator description of the shell theory. In this section we start with the linear shell theory and define linear shell operators with associated bilinear functionals which allow to investigate important properties of the shell operator as symmetry, positivity or positive definiteness. They are of special interest for the construction of numerical approximation procedures like finite element methods. From the relations of section 2 the linear shell equations are obtained by neglecting all non-linear terms. Assuming that the displacements u E H, are elements of a Hilbert-space H,
Buckling and post-buckling analysis of thin elastic shells
199
and that the middle surface forces PE H, ar; elements ofa Hiibert-space H,, we can put them into duality by a bilinear form (u, P) = j-l
M
(w’
(3.1)
+ U&dS
representing the work of the middle surface forces p applied to the displacement field u. Furthermore we define elements CEH, representing the state of strains and BE H, representing the state of stresses of the shell: II
\
NI
NZ2 Ni' &=
’
Ml1
(3.2)
M22
K22 K12
EHe.
+
M’2,
h’zl
The spaces H, and H, can be put into duality by the bilinear form:
<&VU) =
(O,,N;B +u,,M”P)dS JJM
(3.3)
with the linear stress resultant tensor: N;fi
= H”P”‘(j
(3.4)
AlI-
The bilinear form (3.3) is the interaction energy of two linear shell deformations described by E and CJ. We define now an operator F mapping elements ucDI c H, into H, by
J”
=
(3.5)
9
where in the subset DI t H, sufficient differentiability conditions have to be satisfied and rigid body motions are excluded. The constitutive equations of the linear shell theory can be expressed by using an operator
(3.6)
where N;#(O,,) and M@(K~,,) are given by (3.4) and (2.6),: Furthermore an operator .9*: H, + H, is introduced by
y*a=
-
N;fl - -I-(&M”fl + b@M”“) - ;(b;M”fl A 2 Ma”],, + bzp(Nra - b”,M”“)
- bfM”=)
IIP
b;MAP
200
H. STUMPF
which will be used to formulate the linear shell equilibrium equations in operator form. It is shown in equation (3.16) that the operator .Y* is formally adjoint to the operator .j/; The given definitions of linear shell operators and bilinear forms are represented graphically in the following scheme (Fig. 1):
Fig. I
According to Fig. 1 we can define an operator d with the domain D, = Dr and with the range H, by
Nf5(u) - ;(&M"(u) -
+ bf(M”“(u))-
;(&M”“(U)
b$M*“(u)q5 - bJM”BW12 EHP
(3.8)
MoP(~fl,p + b,s(NfP(u) - b”lh4@(u)) which allows one to formulate the linear equilibrium equations of the shell theory in the operator form : &u=p.
(3.9)
In a next step we have to introduce appropriate boundary operators. In the frame of the considered linear and non-linear, respectively, shell theory we define a geometric boundary vector ub representing the three physical components of the displacement vector u and the rotation /IVaround the boundary line C together with the normal displacement ujj at the discrete corner points i = 1, . . . . Correspondingly we define a static boundary vector P, collecting the three components P,, e, P3,the boundary stress couple M,,. and the corner forces Fj at the discrete corner points: PV f: ub
=
,Pb=
U3
PY U3j,j
=
l,...
p3 MVV Fj,j = l,...
1.
(3.10)
We put these geometric and static boundary vectors into duality by the line integral:
[ub,
Pbl
=
,-c @VP,+ u,p,+
u3P3 + PvMvv)ds + 5 u3jF,.
With (3.10) we define the following boundary operators:
j=l
(3.11)
201
Buckling and post-buckling analysis or thin elastic shells
V,UQ
1 I9 t,u=
nu =
UJ -
v3(u3..+ eh)
U,j,j =
1,. . . c
wpw~D-
b”lM”p) - b,,M,,
t=vg Ntp - ;(b”lMAB + b$M”=) - ;(b;M”@ [
‘== <
1
- b,,M,,
- trfM”=)
P. (3.12)
vsMaB(= + $M,.
v=vsMao ,Mtv(S/J + 0) - M,v(s/j - 0), j = 1,. . . , NJ
Corresponding to the operator & of the linear shell equilibrium equations we obtain a boundary operator ti ~=v~(N:~(u) - b$MAP(uj) - b,,M,Ju)
‘=vp Nfa(u) - ;(b”,M”B(u) + b$M”=(u)) - ;(b;M”‘(u) [ au = pZ3-u
=
QM=~I=W +$M,,(u)
- tiM”=(u))
1
- &MA) (3.13)
~=vpMuP(u) Mtv(u)(s/j + 0) - M,v(U)(S/j - O),j = 1,. . . , A’,
which will allow presentation of the static boundary conditions in operator form. The bilinear form (3.3) defines an energy product for linear shell deformations. For a strain field E(U)and a stress field a(v) (3.3) yields the interaction energy of the two shell deformations u and v: (s(u), a(v)) = fJ
mE~=@)NPB(v)+ K=&)MaB(v)ldS ~=,(U)&,(V) + g K=p(Uh,(“)
=
(3.14)
W
1
which, under certain restrictions, allows us to define an energy norm
lll411*=
1
=8aa dS jlrnH [Oaa(u)OA,(u)+ ;K,&)K&I)
=(&~u,.Tu) > 0
for u # 0,
(3.15)
because the operator X’ is positive definite for well-chosen elastic coefficients of (2.5). With (3.1) and (3.14) it can be shown by partial integration that the operator Y*, given by (3.7) is formally adjoint to the operator Y according to (3.5): (Yu,a)
= (u,Y*c)
+ [u,,pa].
(3.16)
It can be proved also that the operator LZ?of the linear shell boundary value problem is symmetric in a formal sense:
202
H.
(d-u, v) = (u, dv)
STUMPF
[p2r~u,v,]
-
(3.17)
+ [p~Yv,u,].
With the operators, introduced in this section, we formulate the linear shell boundary value problem in the following compact form: in M
du=p
KU= u,*
(3.18)
on C,,
au = Pb+ on C, where p are the given middle surface forces, II: the given geometrical boundary datas on C, and Pt the given statical boundary datas on C,. As solution of the boundary value problem (3.18) the displacement field u of the shell middle surface has to be determined.
4.
NON-LINEAR
SHELL
BOUNDARY
VALUE
PROBLEM
IN OPERATOR
DESCRIPTION
For the non-linear elastic shell theory with moderate rotations the middle surface strains yap depend non-linearly on the displacement field u. For further considerations we split the stress resultants Naa in a linear part NfB and a non-linear part N”,B Nap = Nf6 + N”6 m
(4.1)
with N;6
= H”6”‘(j
.Q
(4.2)
With the non-linear part Nip of the stress resultant tensor we introduce a non-linear operator w(u) by defining:
N”.‘(u)-
v?(u) = -
I
+a6(u)N:(u)
-
f(w"(u)N((u)
+ d"(u)Nf(u))
+ $=“(u)N$(u) - Bs’(u)N:(u)
2
II P
-
(4.3 1
b;dW%)
Associated with the operator U(u) we introduce a non-linear boundary operator c(u) given by:
I
v,v6
[Nt6(u)
-
w""(u)N!(u)]
t,v, Nz6(u) - +oa6(u)N:(u) - +“(u)Nf(u)
+ d”(u)N;(u))
+ ;(@=“(u)Nf(u) - @*(u)N’,(u))]
c(u) = <
’
(4.4)
vscp,(u)N”%) 0 (0
J
With the linear operators of section 3 and the non-linear operators (4.3) and (4.4) we formulate the non-linear shell boundary value problem of section 2 in operator description:
203
Buckling and post-buckling analysis of thin elastic shells
du + U(u) = p
in M
KU
= uh*
on C,
au + c(u)
= Pb*
on C,,
(4.5)
where (4.9, are the non-linear equilibrium equations (2.8) expressed in displacements, (4.5), are the geometric boundary conditions (2.13) and (4.5), are the non-linear static boundary conditions (2.14) in displacements. To derive the governing equations of the buckling and post-buckling analysis in a unified and compact form we introduce a non-linear row operator 9’(u) defined by:
9(u) =
du + W(u) in M au + c(u)
(4.6)
au + c(u)
The given middle surface forces p and the given statical boundary contracted to a given row vector F
variables Pt are
(4.7)
Furthermore
we. introduce a bilinear form {u,p) by (&PI = (SP) + [u*rPJ.
(4.8)
It will be shown in the following section that with (4.6)-(4.8) the non-linear shell boundary value problem (4.5) can be given in a variational formulation of the type {.9(u) - F,Q) = 0, ViE A,. 5. VARIATIONAL
PRINCIPLE OF TOTAL POTENTIAL ENERGY LINEAR SHELL BOUNDARY VALUE PROBLEM
FOR THE NON-
As solution of the non-linear shell boundary value problem (4.5) displacement fields Uare obtained, which, as functions of a one-parametric load (4.7), form a non-linear equilibrium path, where points of bifurcation or limit load points can occur. At bifurcation points the fundamental equilibrium path is intersected by a bifurcated path which can be stable or unstable. At limit load points the deformation of the shell structure becomes unstable. In this section a generalized Taylor expansion of the total potential energy of the shell is considered, which leads to a variational formulation of the non-linear shell boundary value problem. Let i denote the displacement field from a configuration ii to an adjacent configuration ii + ti: u=li+Q,
(5.1)
where it is assumed that ii satisfies the geometric boundary conditions (4.5), and that i satisfies the homogeneous geometric boundary conditions a&, = 0 on C,. All quantities will be referred to the known undeformed shell middle surface, which allows to derive the shell equations in a Lagrangian description. For the linearized field variables (2.3) and (2.4)2 ofthe adjacent state we have by superposition:
Inserting (5.2),-(5.2), following form:
into (2.4), the middle surface strain tensor yea is obtained in the
H. STUMPF
204
where yas and )iaDare given by (2.4), with corresponding indications. The total potential energy%,(u) of the moderate rotation shell theory is described by the functional [ 11,121 (P*
%Ju) = J/M Cp[Y&u), k~(u)] - p .u)dS - J
'U +
M,*,,BvIds - j$l FTuaj
(5.4)
cJ
where tE[~,~(u), Q(U)] is the strain energy density per unit area of the undeformed shell middle surface, given by:
(5.5) The first integral of (5.4) is taken over the undeformed shell middle surface M, the second integral along the boundary line C,, and the third term is the contribution ofall corner points of c,. The energy increment A$,(U; i) from a fundamental state ii to an adjacent state ii + ii is a measure for the behaviour of the shell structure at U.It can be derived by using a generalized Taylor expansion of the functional (5.4) leading to the series: A,$,(ii;u) = f&ii + C) - f,(u) = fb(ii;ii)
1 + -fb’(_;u i?) + ;$_.(P;Q”) 2
(5.6) + $s”(U;i”,,
where yb, yi, $ji, 2; denote the first, second, third and fourth Gateaux differentials of the total potential energy /*. They can be expressed in the following form by using the differentials of the strain energy density (5.5): N,
G”‘(ii;ii) - p.ii]dS
&(ii;
i.12)=
II
J”(U;
f;(I& $) = II f;(ij;i*)
-
(P* ‘ti + Mt,,/?,.)ds -
C Ffti,j
j= 1
Cti)dS
M/“(ii ; i.iiXi)dS
(5.7 1
=
All higher order differentials disappear. To calculate the first differential /‘(ii;fi), we have to consider the expression
limG + tii) - Au) 1-O
t
= /‘(U;C)
(5.8)
and if this limit is linear in 8, it is called the first Gateaux differential of the strain energy density Au). Let fi and ti be two different displacement fields superimposed upon the fundamental displacement field ii. The second differential of (5.5) is then defined by: limJ’(ii 1-O
+ tii; ii) - fyi-l; ii) = f”(ii; t
iii).
(5.9)
Differentials of the third and fourth order have to be calculated analogously. Introducing the strain tensor y.&(u)and the change of curvature tensor Q(U) according to (2.4) into (5.5), the differentials of Au) can be determined:
Buckling
and post-buckling
analysis of thin elastic shells
205
1 =P"y&(ii;O) +M=4z=p f'(ii;iii) = 1 (5.10) f(ii; ii) = Ha”“” $j=,ay;,(C;ii) + ;
i?=,k,,
Ha@“’ y:,&i; ii)y;,(U; ii) + j&y;,,(Li, ii) + ; II
k=gZir
j”‘(ir; riiifi) = HaBAr[2y&(ii; ii)y;,,(ti, ii) + y&&i; ii)y$,(i, ii)] f”‘(ii; Uii)
= 3HQBL”y&(ti,ti)yl;,(ii, ii),
where the following differentials of the strain tensor y=#(u) are used:
With (5.7), (5.10) and (5.11) the Taylor expansion (5.6) of the energy increment A%,,(ii;u) in the neighbourhood of a fundamental state ii is given. The principle of total potential energy yields the following statement. For all geometrically admissible displacement fields u, satisfying the geometric boundary conditions (4.5), the total potential energy (5.4) attains its stationary value f;(u;e)
= 0
VikA,;
7tu =
(5.12)
I$
at the solution u = ii. With A,, we denote a Hilbert-space of sufficiently differentiable displacement fields satisfying homogeneous geometric boundary conditions on C,. To derive the Euler-Lagrange equations associated with the variational statement (5.12), we introduce (5.1O)r into (5.7), and transform fb(u;ii) by partial integration leading to the expression : %Ju;i,
= - JI{[
(Nap - ;(b;Mip
- t(w=“NE + cd”Nf)
+ b$M”=) - ;(b;M”O - b!M”=) - fw=BN:
+ ;(Q=‘Nf - fIBAN;)
1
II P
- b;(M”@J, + cpINAP) + p” ti,
+
[(MapI= + cp=Naa)lp + b=,(N=@ - b”,MLB- w=“N{) + p,]li,}dS
[v=va(NaB- b”,M”# - w=‘Nf) - b,,M,, - I’;]&
+ 11C +
[
t=va NaB - ;(b;Mdb (
- ;(w=“Nf: + d”N”,)
+
+ bfM”=) _ ;(b;M”@ + +Nf 2
- @‘N;)
vB(Mn61=+ cp=N=“)+ $ M,, - P:
+
2
+ 0) -
1
1
_ tiMi=) _ fw”N;
1
- b,, M,, - P*, tiI
ii3 + (M,, - M,*,)j$
Mt,(s/j - 0) - F’T]ti,,
ds (5.13)
H.
206
STUMPF
where sufficient differentiability of II is assumed. Introducing (5.13) into the stationarity condition (5.12) yields the non-linear equilibrium equations (2.8) and the non-linear static boundary conditions (2.14) as Euler-Lagrange equations. To derive the Euler-Lagrange equations in the form (4.5) the membrane stress tensor N”” is split into the linear part NTBand the non-linear part N$’ with (4.1) and (4.2). Using the bilinear forms and operators of sections 3 and 4, the first differential of the total potential energy (5.13) can be expressed by: I;(u;ii)
= (SAI + 59(u) - p,C) +
[au+ C(U),&,]” + [cru+ c(u) - Pb*,h], (5.14)
= {Y(u) - F,4j. With (5.14) the stationarity
condition (5.12) is obtained in the compact form:
{.9(u) - F,ir} = 0
vii&,;
7cu
‘d(u) = p
in M
CLU+ c(u) = Pb*
on %,
=
u#f
(5.15)
(5.16)
defining the non-linear elastic shell boundary value problem. In the following sections it will be shown that the equations ofcritical equilibrium, defining snap-through and bifurcation buckling, and also the postbuckling equations can be given in a compact form by using the differentials of the operators d and .d, %%,(I, C, respectively. calculated analogue to (5.8) and (5.9). Because of lack of space they cannot be presented here but are outlined in detail in [16].
6. APPROACH
BY A SEQUENCE
OF LINEAR
SHELL
BOUNDARY
VALUE
PROBLEMS
The total potential energy f&F) is a functional of the displacement field u and of the external load F according to (4.7), where F may depend on various load parameters ij, j = 1, . . . , m. For many stability problems of engineering interest we can restrict our considerations to the study ofshell deformations along a fundamental equilibrium path, described by a oneparameter dependency ~j =
j
nj(~),
=
l,...,m,
(6.1)
where the external load F depends linearly on the load parameter i. F = if
(6.2),
with f denoting a unit load system:
(6.2),
Using (5.12) and (5.15) the stationarity condition for the fundamental path can be given in the two alternative forms: &(ii(A);i)
= 0
{P(ii@)) - Af,iif = 0
ViEt-9”
Vi&,.
(6.3)
We now consider the problem of solving (6.3) in the neighbourhood of a solution point 0 with known displacement field u0 by using the static perturbation technique of Sewell [ 171.
Buckling
and post-buckling
207
analysis of thin elastic shells
Introducing a parameter t with t = 0 at 0, it is assumed that the solution of (6.3) can be expressed by the parametric expansion ii(t) =
Ilo
+
ii(t)
= uo + tlqJ +
t3
t2 -u6’
+ -Ui,”
+
6
2
O(t4)
(6.4)
where a dot indicates differentiation with respect to the parameter t. An expansion of the same type is used to represent the load parameter A(t):
t3
t2 i.(t) = R, + rib + Tib.
+ 6ibb.. + O(t4).
(6.5)
With (6.4) the governing shell operator :f(u(A)) according to (4.6) can be expanded in a Taylor series: .9(ii(t))
=
squ, + i(t))
= !Y(Uo) + tY(u,)u~
+ f
+ ~Y’(uo)ub’]
[w(u&l~
+ ; [9’(U~)U&. + 3.9”(U,)U~Ub.+ ~“‘(UO)U~] + O(t4). (6.6 1 Introducing (6.5) and (6.6) into the stationarity condition (6.3), the following sequential variational statements are obtained: (d(uo) - A”f,iif = 0 - &f, i) = 0 {8’(U&~. + d”(U,)U& - &f,ti) = 0 {W(uo)ub.. + 3Y’(u,)ubub. + d”‘(u,)ub3 - j.b.*f,d) = 0 { d’(u,)ug
vi Vii vi Vi (6.7)
The sequential stationarity conditions (6.7) define a sequence of shell boundary value problems, which allow to determine the unknown coefficients of expansion (6.4). The Euler-Lagrange equations corresponding to (6.7), are the non-linear equilibrium equations and the non-linear static boundary conditions duo + %(uo) = lop.,
au0 + c(uo) = loPb,
in M on C,
(6.8)
with the solution uo. We assume that the solution point 0 is a regular point with a unique solution u. of (6.8). Then (6.7), yields the linear equations: .G/u~ + ~‘(u,)u~ = Ibp, &lb + C’(Uo)U~= &P,*
in M on C,
(6.9)
depending non-linearly on uo. The solution of this linear boundary value problem is the displacement field ug . The variational statement (6.7), leads to the equations .&I~.
+
V’(U,)U~
UUb’+ c’(u&j
=
&p,
-
V”(uo)u&
= &P,* - c”(u,)ui:
in M on C,
(6.10)
208
H. STUMPF
with the solution uv. Correspondingly the variational statements (6.7),, (6.7!,, . . . define linear shell boundary value problems with the solutions ug., ub’.., . . . , the coefficients of expansion (6.4). If this approximation procedure is stopped after n terms the equilibrium error can be calculated, this will not be considered here because of lack of space. 7. CRlTlCAL
POINTS
OF THE
FUNDAMENTAL
PATH
The determination of the critical points of the fundamental path can be performed by considering the behaviour of the total potential energy yJu(t), L(t)) in the vicinity of a solution point uM, &, along every possible one-parameter deformation a(t) u(t) =
u&f +
i(t)
CU’ +
=
i(t)
fu.. +$... +0(t4)
(7.1)
2 = A(t) = AM
superimposed upon the deformation uM, where theload parameter1retains the constant value AM* The energy increment (5.6) at the solution point uy can be obtained as Taylor expansion : A~&,;ii(t))
= ;&‘(u,;u’)
+ f [3~5(u~;uu~.)
+ $&‘(uI(;u~~~.) + fF”(U&&+)]
+ ~~;(u~;u..~)
+ &“(u,;u3)] + ~&‘(u~;u.~u.~)
+ O(?).
(7.2)
It will be shown that at critical points the first term on the RHS of (7.2) will vanish. We have to investigate whether the one-parameter deformation (7.1) can be also an equilibrium deformation, such that the sequential stationarity conditions (6.7) are satisfied for constant I-values. In this case all derivatives of 1 vanish and (6.7) yields:
{!9yuhl)- &#f,ii} = 0 {Y(u,)u.,ii) = 0 (9’(U~)U.. + 9”(U&v2, ii) = 0 {~‘(Uy)U~~~+ 3Y’(uy)uu.. + 8”‘(Uhl)U~3,P) = 0
vii vi vi
(7.3)
vi
Thevariational statement (7.3)r is the stationarity condition for the solution point uM,i., , which is also the trivial solution of (7.3),, (7.3),, . . . for II*= II*+= . . . = 0. If we compare (7.3), with (6.7), it is obvious that (7.3)* has no solution in the non-critical case and therefore also (7.3),, . , . cannot be satisfied. Only in the critical case, denoted by AM = AC,;uhf
and satisfying the stationarity
=
(7.4),
II,
condition {P(uc) - A,f, 6} = 0
vii
(7.4)*
there may be one or more solutions of the variational statement:
{B’(U,)ll~,Q}= 0 Vii
(7.5)
with the restriction that II; has to satisfy some normalizing condition. The index c refers to the critical
point.
Buckling and post-buckling
209
analysis of thin elastic shells
The solution equations of (7.5) yield the conditions for critical equilibrium, the equations of snap-through or bifurcation buckling of shells: du; + ~‘(u~)u; = 0 au; + c’(u,)u; = 0
in M on C,
(7.6)
with the linear operators & and a, defined by (3.8) and (3.13). The operators U’(u,) and c’(u,) are the Ghteaux derivatives of (4.3) and (4.4) and depend non-linearly on the prebuckling deformation u,. The linear equation (7.6), represents three homogeneous field equations of the shell middle surface and (7.6), the associated static boundary conditions. They define a homogeneous eigenvalue problem depending non-linearly on u,(&). The solution of the eigenvalue problem (7.5) and (7.6), respectively, yields the buckling mode u; with the associated buckling load I,. To normalize the buckling mode u; we choose a normalization according to the strain energy of linear shell deformations by setting: {Y’(o)u;,u~> = 1.
(7.7)
Here Y(0) denotes the derivative of the operator P(u) at u = 0. Using definition (4.6) of the operator Y(u) and definition (4.8) of the bilinear form {. , .}, expression (7.7) represents the double elastic shell energy for the small deformation u;: {d’(O)u;,u~) = (du;,u;)
+
[au;,IQ,] O,&)OA,(u;) + ; K~,#;)K&)
1
dS.
(7.8)
Using u; to define shell buckling modes, higher order contributions ofexpansion (7.1), are not considered and the question, whether or not (7.3),, . . . are satisfied, is not discussed. If we choose in (7.5) ii = U;E& we can prove easily: {Y’(u,)ll;,
u;}= y;(u,;u;2) = 0,
(7.9)
leading to the result that the first term of the energy expansion (7.2) vanishes at uc, I,. This corresponds to the energy criterion of stability that for critical deformations the second differential of the total potential energy must disappear. If we compare the stability equations (7.6) with the governing equations (4.5), and (4.5), of the non-linear shell boundary value problem, it is obvious that (7.6) is the differential of (4.5), and (4.5), for A(t) = A,. This gives a very general form of the so-called adjacent equilibrium criterion well-known in the engineering literature [ 181. 8. BIFURCATION
AND
POST-BUCKLING
ANALYSIS
Let us assume that, at 3, = AC,, a new equilibrium path ub(A)bifurcates from the fundamental path ii(A): l+(A) = ii(A) + v(A).
(8.1)
With v(A)we denote the differential displacement field from the fundamental to the bifurcated path. Under suitable regularity conditions the fundamental equilibrium path ii(A) can be expressed by the Taylor expansion near the bifurcation point u,, A,: ii(l)
=
UC +
(A -
;(A- It,,‘lz””
n,)uup+
+
;(A- ~c)3~ooo+ W
- A,r),
(8.2)
where the symbol ( )’ indicates differentiation with respect to the load parameter I, ( )’ d( ) The index c refers to the bifurcation point. = =. L
210
H. STUMPF
For the differential displacement v(A(t)) we use the parametric representation at the bifurcation point: t2 t3 v(t) = tv; + TV;.’ + gv;.
The corresponding
. +0(P).
with (t = 0)
(8.3)
expansion of the load parameter l(t) near t = 0 yields: l(t)
= 2, + tl; + GA;.
+ :A;.+
+ O(t”).
(8.4)
The coefficients of the parameter expansion (8.3) may be normalized according to the bilinear form (+“(O). , .) : {Y(O)v;,v;}
= 1
(8.5)
{Y’(O)v;, vi.1 = {9’(O)v;, Vi..} = . . . = 0.
Assuming that the bifurcated path u”(i) is an equilibrium path the stationarity condition (~5.3)~must be satisfied: (@)
- lf,ii} = 0
vii
(8.6)
With (8.1) we use the Taylor expansion of the operator Y@“(i)): .?P(Uh(;L))= 9(U) + Y’(ti)v
yielding the two stationarity
+
;~t’(u)v2 ++(u)v’
(8.7)
conditions: VI-i
(9yi-i) - J&f,ii} = 0
(8.8 { P’(U)v + ;Y(ii)v2
+ ++‘“(ti)v’, ti) = 0
vi
Relation (8.8), is the stationarity condition for the fundamental path and (8.8), the stationarity condition to define the bifurcated path. For further considerations it is presupposed that the solution of (8.8),, the fundamental path (8.2), is known. The operators of (8,8)2,9’(U), Y’(U) and Y”(ii) will be expanded at the bifurcation point. With (8.2) and (8.4) we obtain the series: 9’(ii(i(t)))
= W(u,) + tiLy’(U,)up+ +
[~~2w’(u,)upo+ i~;GP”(u,)up
;
~;29”‘(u,)up2] + o(t3)
P”(ii(A(t)))
= P"(u,)
+ ti.~"'(u,)u~+
d”‘(ii(A(t)))
= W”(U,) = 9”‘.
(8.9)
O(t2)
We introduce the operators (8.9) into the stationary condition (8.8)2. Replacing v by (8.3) and collecting the various terms with respect to the powers oft, the following result can be derived : Vi
{Y(u,)v;, ii} = 0 {Y’(uc)v;. + P”(u,)v;2 + 2~;~“(u,)upv~,o)
v’e
= 0
{ Y’(u,)v;~ . + 3~“(u,)v;v;~ + Y”(llJV;3 + 31;[~“(u,)upv;*
+ Y”‘(Ur)U;V;2]
+ 32;2 [.V’(uC)up”v~ + B”‘(u,)up2v;] + 31;~B”(u,)upv;, ti) = 0 vi
(8.10)
Buckling
and post-buckling
analysis of thin elastic shells
211
representing a sequence of stationarity conditions to determine the bifurcated path (8.1) The first variational statement (8.1O)r describes together with the normalization condition (8.5), the homogeneous eigenvalue problem of bifurcation buckling of thin shells. With definition (4.8) of the bilinear form {. , .} and with the operator 9’(u,) the solution equations of (8.10), dv; aV;
+ U’(u,)v; = 0 +
e’(U,)V;
=
0
in M
(8.11)
on C,
are the stability equations, depending non-linearly on the prebuckling deformation II,(&). Equation (8.1 I), represents the field equations and (8.1 1)1 the associated static boundary conditions. The solution of the eigenvalue problem (8.11) yields the buckling mode v; together with the bifurcation load parameter A,. In this section it is assumed that ACis a singular eigenvalue. Because of (8.1) the increment of the bifurcated deformation at the bifurcation point is: Ub. =
u; + v;.
(8.12)
To determine the coefficient rl; of series (8.4), we choose in (8.10)2 ii = v;E&. symmetry of the operator F(u,) [16] we obtain with (8.10),: {9’(ll$;~,
Vi} = {v;.,B’(u,)v;;
= 0
Using the
(8.13)
Then (8.10)2 yields for ii = v;: *. = c
_A
P”(~)v~2~ Vi)
2 {s”“(u,)tJpv;, v;}’
(8.14)
where it is assumed that {B”(u,)u~v;.,v;} # 0. With known v; and rl; (8.lO)z is a variational statement to determine the coefficient v;.. The associated solution equations J&Q;*+ %?‘(uc)v;*= - ~“(uc)v;2 - 2A;V’(u,)upv; (8.15) nv;. + G’(U,)V;’= -c”(llJV;~ - 2i;c”(t4)$%; form a set of linear nonhomogeneous partial differential equations, describing a linear shell boundary value problem yielding the displacement vi*. The first line in (8.15) represents the field equations, the second line the associated static boundary conditions. The solution of (8.15) is not unique, because any multiple of v, can be added to a particular solution of (8.15), which we denote by vi;. To satisfy the normalizing condition (8.5), the unique solution v;* is given by: Vi’
=
v;; - { B’(O)v;;,
v;}v;,
(8.16)
which can be shown by introducing (8.16) into (8.5),. Continuing the procedure we choose in (8.10)3 i = v;~fi, such that {~‘(u,)v;~.,v;~ disappears because of the symmetry of Y(u,) and because of (8.1O)r. Then the load coefficient 1;. can be determined from (8.10)3 yielding:
{a”(U,)V;V;.’ +
),; . =
-
+
f.s~~~(uJY;” +&[9”(u,)upv~* + Y”‘(u,)upv;2] Ai2[w’(uJup” v; + ~“‘(uc)llp2v;],Vif {3p”(u,)upv;,vi;
(8.17)
H. STUMPF
212
For many shell buckling problems the bifurcation is symmetric. In this case the coefficient 1; of the load parameter expansion (8.4) vanishes leading to a considerably simplified expression for I; .:
{.~“(u,)v;v;~+ A;. =
-
f9’.‘(uc)v;J. v;.)
{Y’(u,)upv;,v;}
.
(8.18)
With known A;, A;-, v;, v;. expression (8.lO)3 is a variational statement to determine the displacement field v; . .. The associated solution equations define a linear shell boundary value problem, which allows one to determine vi**.Analogously the further variational statements of (8.10) yield the higher order terms of the series (8.3) and (8.4) describing the bifurcated equilibrium path. With the load coefficient (8.14) of the parameter expansion (8.4) we obtain a necessary condition for stability of the shell at bifurcation points { P”(U,)Vf2,Vi} = 0
(8.19)
or, equivalently, 1; = 0. A further necessary condition for stability is {.GY”‘(u,)v;3, v;} - 3{8’(u,)v;~,v;~} > 0,
(8.20)
for which the increment of total potential energy Afp(uc;ub) is positive. The homogeneous eigenvalue problem (8.11) of shell buckling may have n linearly independent solutions, the eigenmodes v;(~),i = 1, . . . , n associated with a multiple buckling load d,. In this case we have to modify slightly the post-buckling analysis of this section by introducing a multiple buckling mode [16]. 9. NUMERICAL EXAMPLE Let us consider a circular cylindrical shallow shell panel with two opposite edges hinged and two opposite edges free (Fig. 2). Using a finite element procedure this example had been calculated in [19] for a central point force P and in [20] for a combined loading of central point force P and axial compression N under the assumption of a fixed ration N/P. This assumption leads to a one-parameter deformation represented by an equilibrium path. Here we want to investigate the more general case of a two-parameter dependency such that P and N can be varied independently. Furthermore an imperfection or eccentricity E of the point load P is introduced (Fig. 2): The geometry of the shell panel with constant thickness h, radius R, length 2L and opening angle cpo can be described by two shell parameters
Fig. 2. Circular
cylindrical
shell panel with eccentric load P and axial load N.
Buckling and post-buckling analysis of thin elastic shells
213
= R&/h and p2 = RqdL. With IA,v, w, we denote the axial, circumferential and normal displacement components in nondimensional form. The displacement field (a, v, w) has to satisfy the geometric boundary conditions at the hinged edges: v = w = 0. Using only nondimensional quantities the total potential energy of shells according to (5.4) leads for the shell problem of Fig. 2 to a functional f = f(u, v, w, pl, p2, P, N, E) depending on the displacement components u, v, w, two shell parameters cc,, p2, two load parameters P, N and the eccentricity E. To determine the stability boundaries of this shell structure a Rayleigh-Ritz approximation can be used by choosing a set of geometrically admissible displacement fields with unknown coefficients, complete in a well-defined space. For a first approximation we choose Pl
u(x) = -Usinix
v(q) = Vsin ncp
(9.1)
w(x,q) = W, cosicp + W,sin7rq
with the unknown coefficients U, V, W,, W,. Assumption (9.1) satisfies the geometric boundary conditions at x = x/L and cp = (p/q,. Introducing (9.1) into the total potential energy (5.4) leads to a functional %
=
f(u,
K
w,,
K
PI,
~2,
P,
N,
~1.
(9.2)
Differentiation with respect to the unknown coefficients yields four equilibrium equations: 8%
-
au =
0;: =o;g=o;g=
0.
s
(9.3)
Y
Equations (9.3), and (9.3), are linear in U and V, respectively. Eliminating U and V and introducing them into equations (9.3), and (9.3)., gives two non-linear equilibrium equations in the unknown coefficients W, and W,. The snap-through buckling of the cylindrical shell panel will be governed by symmetric deformations. Therefore we can choose the coefficient for unsymmetric deformation W, = 0 and obtain one equilibrium equation: W,3e3 + W,2e2 + W,e, + e. = 0
(9*4)1
with: eo=-
128v/n3
N --
2
p.,
e2 = - 8.737
Ccl
2.029” 2 4 e, = 3.043 + 2 -2-N; Pl
Using the transformation
(9.4)~ e3 = 5.644
Pl
W, = W - $
and putting 3
(9.5) yields an equilibrium surface defined by:
M = {(w, f0,fl)lW3 -f1W+fo
= 0).
(9.6)
214
H. STUMPF
The critical points of the equilibrium surface (9.6) are the snap-through buckling points, for which the second differential of the total potential energy vanishes according to (7.9) yielding:
Expression (9.7) represents the collection of critical snap-through buckling points for a twoparameter loading and corresponds to a cusp-catastrophe [15]. The geometrical representation is given in Fig. 3 showing the collection of upper and lower snap-through buckling loads as well as the areas of stable and unstable behaviour of the shell. Figure 4 shows the equilibrium surface (9.6) in the three-dimensional (W,, P, P/)-space and the collection of critical snap-through buckling loads in the (P, N)-plane. Furthermore we have to investigate the unsymmetric bifurcation buckling by taking into consideration unsymmetric deformations with W, # 0, E # 0. It can be shown that for special values of axial compression N, bifurcation buckling exists and the collection of critical bifurcation buckling loads can be determined analogue to the results for symmetric snapthrough buckling. CUSP - CATASTROPHE
lower
critical
Fig. 3. Cusp catastrophe
of snap-through
Equilibrium
/N
Fig. 4. Equilibrium
buckling for two-parameter
loading
Surface /
/
Collection ot Critical
surface and cusp catastrophe
Points or snap-through
buckling
({I, = 1)
Buckling and post-buckling analysis of thin elastic shells
215
Arkno*,le~gements-Theauthor would like to thank the Polish Academy ofSciences inGdadsk for partial support of this work and Prof. Dr. W. Pietraszkiewicz for stimulating discussions. REFERENCES 1. W. T. Koiter, On the stability of elastic equilibrium, Ph.D. Thesis, Polytechnic Institute. Dclft (1945). NASA (translation) TTF-IO, 833 (1967). 2. W. T. Koiter, Elastic stability and post-buckling behavior. In Proceedings of rhe Symposium on Non-linrur Problems, (Edited by R. E. Langer), pp. 257-275. University of Wisconsin Press (1963). 3. W. T. Koiter, General equations of elastic stability for thin shells.In Proceedings ojthe Symposium on Theory oj Shells to Honour Lloyd H. Donnell, pp. 187-227. University of Houston (1966). 4. B. Budiansky, Theory of buckling and post-buckling bchaviour of elastic structures, Adu. appl. Mech. 14, 1-65 (1974).
5. W. Pietraszkiewicx, Introduction to the non-linear theory of shells, Mirr. Inst. Mech. 10, Ruhr-Univ. Bochum (1977). 6. 1. L. Sanders, Non-linear theories or thin shells, Q. uppl. Math. 21, 21-36 (1963). 7. W. T. Koiter, On the non-linear theory of thin elastic shells, Proc. Kon. Ned. A&. Wet., Ser. B, 69, 1-54 (1966). 8. L. P. Nolte, Beitragzur Herleitung und vergleichende Untersuchunggeometrisch nichtlinearer Schalentheorien unter Berticksichtigung groBer Rotationen, Mitt. Inst. Mech. 39. Ruhr-Univ. Bochum (1983). 9. W. Pietraszkiewicz, On consistent approximations in the geometrrcally non-linear theory ol’shells, Mirf. In~r. Mech. 26, Ruhr-Univ. Bochum (1981). 10. L.-P. Nolte and H. Stumpf, Energy-consistent large rotation shell theories in Lagrangean description. Mech. Rex Comm. 10.213-221 (1983). Il. H. Stumpf, The derivation of dual extrcmum’and complementary stationary principles in geometrical nonlinear shell theory, Irrg. Arch. 48, 221-237 (1979). 12. R. Schmidt and W. PietraszkiewicG Variational principles in the geometrically non-linear theory of shells undergoing moderate rotations, Ing. Arch. 50, 187-201 (198 1). 13. H. Stumpf. On the linear and non-linear stability analysis in the theory of thin elastic shells, Lecture XV ICTAM Toronto/Canada 1980, Ing. Arch. 51, 195-213 (1981). 14. H. Stumpf, The stability equations ofthe consistent non-linear elastic shell theory with moderate-rotations. In I UTAM-Symposium: Srabiliry in the Mechanics oj Coarintta, Niimbrecht/Germany 1981, Proc. 89- 100. Springer, Berlin (1982). Srahilify und Morpliogcnesis. Benjamin, Reading, MA (1975). 15. R. Thorn, Swwrural 16. H. StumpL Undied operator description, non-linear buckling and post-buckling analysis ofthin elastic shells. Mitt. Inst. Mech. 34, Ruhr-Univ. Bochum (1982). 17. M. J. Sewell, A general theory of equilibrium paths through critical points, 1, II. Proc. R. Sot. A306,201-223, 225-238 (1968). 18. D. 0. Brush and B. 0. Almroth, Buckling of Bars, P/aces and Shells. McGraw-Hill, New Yqrk 1975. 19. A. B. Sabir and A. C. Lock, The application of finite elements to the large deflection geometrically non-linear behaviour of cylindrical shells. In Variationul Methods in Engineering, (Edited by C. A. Brebbia and H. Tottenham), 7/54-7/65. Southampton, University Press, Southampton (1972). 20. E. Stein, A. Berg and W. Wagner, Different levelsofnon-linear shell theory in finite element stability analysis, In Buckling CI/ Shells. (Edited by E. Ramm), Springer. Berlin (1982).
Zusarmrenfassung: In der vorliegenden Arbeit wird die nichtlineare Stabilitatsund Nachbeulberechnung elastischer Strukturen im Ftahmen einer geometrisch-nicht-linearen Schalentheorte mit moderaten Rotationen untersucht. Unter Benutrung einer vollstandig Lagrangeschen Betrachtungsweise werden Variationsaussagen sowie die rugehorlgen Systeme nichtlinearer Schalenglelchungen unter ElnschluB der Randbedingungen hergeleitet. die eine Berechnung des fundamentalen Gleichgewichtspfades, kritischer Punkte mit Durchschlagsoder Verzweigungsbeulen sowie der NachbeuldeUrn die erhaltenen Glelchungen formationen ermoglicben. In einer kompakten Form darstellen zu konnen, wird eine Dlese verelnfachende Operator-schreibweise eingefuhrt. ermoglicht daruber hinaus den Nachweis elniger msentlicher Eigenschaften, die zur Ronstruktion.von Approximationsverfahren wie Finite-Element Methoden von Bedeutung sind. Ein Beispiel wird unter Ritz Ansatzes numerlsch fur eine Zmi-Parameter Durchschlagspunkte eine
Benutrung eines einfachen Rayleighuntersucht. Es wird gezeigt, da8 Belastung die Henge der kritischen Ketastrophe vom Typ “Cusp” ist.